Applications of the generalized Sylvester matrix

Applications of the Generalized Sylvester Matrix J. Maroulas

and D. Dascalopoulos

Department of Mathematics National Technical University of Athens Athens, Greece

Transmitted by Robert Kalaba

ABSTRACT Recently a Sylvester matrix for several polynomials has been defined, establishing the relative primeness and the greatest common divisor of polynomials. Using this matrix, we perform qualitative analysis of several polynomials regarding the inners, the bigradients, Trudi’s theorem, and the connection of inners and the Schur complement. Also it is shown how the regular greatest common divisor of m + 1 (m > 1) polynomial matrices can be determined.

1

INTRODUCTION Consider

a set of m+ 1 polynomials

b,(A)=bi,A”

+&A”-’

+ . . . +b,,

(i=O,l,...,m)

(1.1)

over some field, where b,(A) is manic (b, = l), and pdn is the maximum degree of the remaining polynomials b,(A), . . . , b,(A), i.e., bi, “_p #0 for at m, j
b(h)= i i&(A)

(1.2)

i=l

foranyvectork=[lc,,..., k,] divisor (g.c.d.) of polynomials

different from zero. Thus, the greatest common in (1.1) is also the g.c.d. of b,(X) and b(h).

APPLIED MATHEMATICS AND COMPUTATION 8:121-135 (1981)

121

0 Elsevier North Holland, Inc., 1981 52 Vanderbilt Ave., New York, NY 10017

00QWO03/81/020121+

15$02.50

122

J. MAROULAS AND D. DASCALOPOULOS

Recently, an extended Sylvester matrix R, for the set of polynomials b,(h), i=O,l,..., rn, has been defined [7, 51:

Ro R,=

Rl .

(1.3)

3

_d,_ where ... . . .

R,=

0

l.

is a pX(n+

.

b01

-.-

P ) matrix and

bi,n--p

...

bi,n-l

bin

0 Ri= 0

bi.n-p

bi,n--p bi,n--p+l .. i=l

bi,n-l bi*

bi,n

0

0

,**.> m,

are n X ( n + p ) matrices. Obviously, for m=l, R, is the classical Sylvester matrix. In this paper, first introducing the ideas of inners and bigradients by the matrix R,, the determination of g.c.d. of b,(X) is undertaken, generalizing Trudi’s theorem (Theorem 1). In Sec. 3, extending a previous result [6], it is shown (Theorem 2) that the rank of inners of R, is equal to the rank of suitably determined matrices of the Schur complement of R,. Also, it is expressed for outers instead of inners. In Sec. 4 the regular g.c.d. (if it exists) of several polynomial matrices is determined, extending directly a method for two polynomial matrices [2]. Next, in Sec. 5, it is shown how the results of Sets. 2 and 4 are carried over for the generalized polynomials with respect to an arbitrary polynomial basis [14]. Moreover it should be noted that the ideas involved in this paper are closely related to problems in linear control system theory [9, lo].

123

Applications of Generalized Sylvester Matrix 2

INNERS Denoting

AND BIGRADIENTS

OF SEVERAL

POLYNOMIALS

by

ci=

$ kibii

(j=n-p,n--p+1,...,fl)

i=l

the coefficients

of b(X) in (1.2), and by c”_p

R=

1

%I Ra ’ [

0

_:

**.

*:

C”_l

c,

*:

R,= 0

c”-p

C”-p

-.*

cn-p+1

-*.

C”_l

crl

C”

0

0

(2.1) the Sylvester matrix, of order n + p, of the polynomials b,(A), b(h), it is easily verified that the matrices in (1.3) and (2.1) are related by the equation

R=QR,,

(2.2)

where

and @ denotes the Kronecker product. Therefore, since the matrix Q has full rank, from (2.2), in a rather simple way, it is recovered [5] that rank R s = rank R =n+p4,

(2.3)

where 8 is the degree of the g.c.d. d(X) of b,(X). Furthermore, letting B,( be, b, ), B,( be, b) be the matrices which are formed from R,, R, respectively, by deleting the first t rows and columns from all sides, then (2.2) implies

‘,(b”,b)=Q,‘,(b,,bi),

(2.4)

124

J. MAROULAS AND D. DASCALOPOULOS

where Q, is formed from Q in the same way. So, from (2.4) for any k ( #O>, follows that rankB,(bO,b)=rankB,(bO,bi). Moreover,

(2.5)

of I?,( b,, bi) by

if we replace the last column

hP-‘-‘b,(X),...,

it

Ab,(X),b,(A),b,(A),Xb,(X),...,A”-’b,(A),..., X-‘b,_,(h),

b,-,O),...,

b,(h),

Ah,(X),...,

X-‘-‘b,(h),

the following matrix I?,( b,, b,, h), due to (2.4), is related with the well-known bigradient B,(b,, b, X), whose last column is [8] hP-t-‘b,(X),...,

Xb,,(X), b,(X),

b(h),Xb(A)

,..,,

X-+‘b(h),

by the equation

(2.6) Denoting by C,(b,, bi, A) the submatrix rows corresponding to the zero columns AP-t-lbo(A),...,

Xb,(h),

b,(A),

of B&b,, bi, X) after eliminating of Qt -i.e., its last column is

b,(A),Xb,(h)

,...,

F-‘b,(X)

b,@),..., -the

simplified

formula

,...,

An-t-lb,(h),

of (2.6) is

B,(b,, b, A)=

1

zp-t kB;_ C,(h,JiJ)~ o

n

from which evidently

the

t

we obtain

detB,(b,,b,h)=

2

kik,...k,detAi,i

i,j,...,/

,,,,, I,

(2.8)

of Generalized SylvesterMatrix

Applications

125

where ...

...

...

b,,

1

b0,n+p_-2t_-2

XP-‘-‘b,(X)‘

1 %,(X) ...

bo,,-t-1

b,(X)

bi,n-p

. . .

bi,p-*--l

b,(X)

...

.a.

bi,p-t

X$(A)

-. .

-. .

bl,.+p_-2f_-2

A”-‘-‘bl(X)

1

bi,n_p

1bl,n_p

...

. +.

(2.9)

i,.

and the summation is over all arrays (i, . . ,I) by repetition, up to n-t times, of the numbers { 1,. . . , m}. Hence, from the equations (2.5) and (2.8), for any k, we derive Trudi’s result [8] for several polynomials:

THEOREM 1.

Zf rankB,(ba,

b,)tn+p-2t

S-l, and the matrix B,(b,, bi) has fullrank, then the g.c.d. of b,(X) (i=O,l,..., m) has degree 6, and it is equal to det Ai, i ,,,,, , for any array (i,j,...,E) as above. Conversely, if the g.c.d. of b,(X) (i=O, 1,. . . , m) has degree 6, then it is given by the determinant (2.9) for any array (iTi,..., l), and Bs( b,, bi) is the maximum dimension matrix with full rank.

for t=o,1,...,

From the above theorem it is clear that the matrices B,( b,, b, ), B,( b,, bj, A) extend the ideas of “inners” and “bigradients” for more than two polynomials. Also, this terminology is applied to the matrices C,(b,, hi), C,( b,, bi, A), where C,( b,, bi) is formed from Bt( b,, hi) like the abovementioned Ct(b,, bi, A)* EXAMPLE. Consider

the polynomials

b,(X)=A4+2A3-X2-2A

[=X(X2-l)(h+2)],

b,(X)=

[G -(h+2)(&3)(A+l)]

-A3+7h+6

b2(X)=2A2+6X+4

[E2(X+l)(A+2)].

’

126

J. MAROULAS AND D. DASCALOPOULOS

Here p = 3, and the generalized 1

2

0

:--------1

Sylvester -1

matrix is -2

2

:I-

r----------i

1 . . . 0’..)............................(... 0, 0 I 01

2

0’

0

-l

0

o-

0:

0 ’ 0

-2

I

0; 7 I 6 0 7 6 1 0 O I -11 0 7 6 i 0 ; 0 . .-1,. . . . . . .0;. . . . . .7 . . . . . 6. . . . . . 0’ . ..I......... 0,o

R,=

0; 0,-l

0 ----- --2

-1

01 0 0 0 ’ L_O___H___!, L_!_-_2____6____4___!_1

0

2

6

I ’

2

6 ’ 4 4’0 ’ 0

4

0

0

0

Since rank R, ~5, (2.3) implies 6~2. Also, the degree of the g.c.d. can be found from the relations rank $( be, bi ) < 7, rank B,( b,, bi ) < 5, and rankBs(ba, hi)=3, where B,(b,, bi) (1=1,2) are the matrices within the first and the second dashed lines. Furthermore, if we replace the last column of &(‘a,

hi) by

then according to Theorem of the determinants 1 0 -1

2

b,(X)

0

b,(h) hb,(A)

-1

1 the g.c.d of b,(X), b,(h),

I ,

0 0

2

WV

2

b,(X) hb,(A)

-1

,

-1 It is worth polynomials divisions of expressions

b,(h)

is given by any

I

2

b,(X)

0 0

0 2

b,vd hb,(A)

1

2

b,(X)

0

0

b,(X)

0

Xb,(X)

1

.

noting that this procedure of determining the g.c.d. of several is rather preferable to the Routh array or the Routh array without b,(h), b(X) [4], since the elements of the tableau will be algebraic in k 1, . . . , km and a special effort is needed.

Applications of Generalizd

3.

AN EXTENSION Consider

127

Sylvester Matrix

OF THE SCHUR

the partitioning

P and the corresponding

COMPLEMENT

of Sylvester

matrices

n

P

*

Schur complements (R/R,)=R,

-R,R,‘R,,

(Rs/R>)=R;

-RjR;-‘R;,

which, due to (2.5) are easily recognized

to be connected

by the equation

(R/R,)=(k~ZI,)(R,/R’,).

(3-I)

If 0

1

0

0

0

1

: 0

0

0

_ -4lrl

-boJ-l

c=

is the companion

matrix of b,(X),

0 *

h,n-2

...

1 +“I_

then from (3.1) and the relationships

1

b,(C) b(C)=(k@Z,)

f i b,,(C)

and [l] (WR,)=WW it can be established

(3 *2)

(3.3)

that

(3.4)

J. MAROULAS AND D. DASCALOPOULOS

128

which seems to be an extension of (3.3). Symbolizing by Xct) the matrix produced from X after deleting the last t columns and rows, then from (3.2) we have b,( C)‘t’ :

b(C)‘t’=(k@Z,_,) i which, for any k (#O),

I ,

b,,,(C)@’

implies

.

Moreover,

using the relationship

(3.5)

[6]

det b(C)‘t’=det

R,(b,,

b)

(3.6j

and (2.5), (3.5), and (3.6), we have

rank I?,( b, , bi) = rank

So the content

of (3.7) is expressed

(3.7)

by

THEOREM 2. TIhe rank of the inners of R, is equal to the rank of the block column matrices with elements the matrices formed by deleting the appropriate last rows and columns of b,(C).

Denoting

by A,( b,, bi ) the outers [4] of

129

Applications of Generalized Sylvester Matrix

i.e. the matrix which is built up by combining portions from each corner, as follows:

p--t

p--t

Then due to

it is evident that in Theorem 2 the inners of R, can be replaced by the outers of s. Furthermore, if ri are the columns of b(C) and

is the g.c.d. of the polynomials b,(A), b(X), then from (3.2) and the equations

ri =d s+l_ira+l+

i xiii; j=6+2

(i=1,...,6)

the known relationships [l] are recovered.

C~ROLLAM

1. The last n-8

are linearly independent

columns ri’ of

and satisfy

the equations

(3.8).

(3.8)

130 4.

1. MAROULAS AND D. DASCALOPOULOS THE G.C.D.

OF SEVERAL

POLYNOMIAL

MATRICES

Letting P(X) be a regular polynomial matrix of order 1 and degree n, and Qi( A ) a set of m polynomial matrices of dimensions o X 1 and degree at most n- 1, without loss of generality we can write P(h)=Zh”+A,~n-’

+ . . . +A,_,h+A,

Qi(h)=Bi,X”-‘+

. . * +Bi,,_$+Bi

n

(i=l,...,?n)

(4.1)

where the elements of matrices are in some field. Denoting by fl, QCx)=

2

kiQi(X)

i=l

=Qi$-’

+

. . +Q,-lh+Q,,

where (j=l,...,n),

2 kiBii

Qi=

i=l

then due to

I[

P(A)

Q(X) = for any k (#O)

zr

0 k@Z,

0

we have:

THEOREM 3. The polynomial matrices P(A), Q,(A) prime if and only if

P(A)

Q1(x)

rank

~1

_

QmiW

_

for all A.

are relatively

right

Applications

131

of Cenerulized SylvesterMatrix

The approach

to the determination

of the right g.c.d.

D(X)=ZP

+ . . . +DG_lh+DS,

of order 1, of P(A), Q,(h) on the following:

+qP’

(4.2)

(if it exists) is based, as for the scalar polynomials,

LEMMA 1. The polynomial matrices P(X), Q,(h), . . . , Q,(h) have a right common divisor (r.c.d) if and only if, for any vector k (#O), it is a r.c.d. of P(X),

Q(h).

PROOF. A r.c.d. of P(X), Q(A) for any k (# 0) is also a r.c.d. of P( A ), Qi( h) (i-l,..., m), taking k= ei, where e, is the row vector with one in the i-th component and zero elsewhere. W It follows from this lemma that the right g.c.d. D(A) of P(X), Qi(X) is also the right g.c.d. of P(X), Q(h) and vice versa. Denoting by

1,

0

0

4

-A,-1 the block companion

matrix of P(h),

-4-2

0 ... -..

1, -A1

1

and

Bi

B,C

u(Bi,C)=

where

we have:

(i=l,...,m),

(4.3)

132 THEOREM

J. MAROULAS AND D. DASCALOPOULOS 4.

The degree 6 of D( A) is given by the formula

WLC) 6=n-

(4.4)

frank

U(B,,C) : I. PROOF.

Let

and

QO QOC

@(Q”TC)=

(4.5) QOCn'-l

From the equation

,

it is implied

that the matrices

in (4.3) and (4.5) satisfy the relationship

cP(Qo,C)=(k@Z,,@Z,)M

w%J3 ;

I I >

U@,,C)

where M is a suitable

permutation

matrix. Thus, due to [2], to

(4.6)

133

Applications of Generalized Sylvester Matrix

and to the fact that the matrix (kG3Z,,@Z,)M from (4.6) we obtain (4.4).

COROLLARY

2.

ww) U((R,, :C) 1

The polynomial

and only if the matrix

I has full rank.

is nonsingular for any k (#O), n

matrices

in (4.1) are relatively

prime

if

For the determination of coefficients Di of D(h), consider the matrices F(Qa, C),V(Bi, C) which are formed by the first (n- 6)Zv rows of @(QO, C), U( B,, C) respectively. From (4.6) it can easily be established that

I 1 V@,,C)

F(Q,,C)=(k~Z~,_,,,~Z”)~

:

V(R,,

>

(4.7)

C)

where fi is another permutation matrix. If F,, T, (i = 1,. . . , n) are the block columns of

respectively, of dimensions (n-

S)lv X 1, then (4.7) implies (i=l,...,m).

F, =(k@Z~,_,,,G3Z,)tiT,

(4.8)

Thus, using the result [2] that the matrices F6+ 1,. . . , F, are linearly independent and F, =F8+lD8_i+,+

i

qXii

(i=l,...,fS),

i=6+2

then from (4.8) and (4.9) for any vector k( #O),

we obtain:

(4.9)

J. MAROULAS AND D. DASCALOPOULOS

134 THEOREM 5.

The

block co1umn.s Ts+l,. . . , T, are linearly

independent,

and if

q=

qxii

i j=6+1

then

5.

THE CASE OF GENERALIZED

POLYNOMIALS

Let the polynomials b,(X) in (1.1) be given in generalized expressed as a linear combination of the polynomial basis

defined

by the recurrence

form [7], i.e. be

relationships

fi‘(‘)= I: (‘ijh+Eij)fi-l(h)T

i>l.

j=l

Using an analogue technique as in [ 131, then a generalization of Theorem 1 is obtained and the g.c.d. of b,(X) in generalized form is determined without expanding them into power series. Similarly, Theorem 5 is extended in the case where the polynomial matrices (4.1) are in generalized form. REFERENCES S. Barnett, Greatest common divisor of several polynomials, Proc. Cambridge Philos. Sot. 70:263-268 (1971). S. Barnett, Regular greatest common divisor of two polynomial matrices, Proc. Cambridge Philos. Sot. 72:161-165 (1972). S. Bamett, Interchangeability of the Routh and Jury tabular algorithms for linear system zero location, in Proceedings of the IEEE Decision and Control Confwence, 1973, pp. 308-314.

Applications 4 5 6 7 8 9 10 11 12 13 14 15 16 17

of Generalized Sylvester Matrix

135

S. Barnett, Some properties of inners and outers, in Proceedings of the ZEEE Decision and Control Conference, 1978, pp. 836-900. S. Barnett, Greatest common divisors from generalized Sylvester resultant matrices, Linear and M&linear Algebra, to appear. S. Bamett and E. I. Jury, Inners and Schur complements, Linear Algebra and A&. 2257-63 (1978). E. W. Cheney, Zntroduction to Approximation Theory, McGraw-Hill, New York, 1966. A. S. Householder, Bigradients and the problem of Routh and Hurwitz, SZAM Reu. 105666 (1966). E. I. Jury, “Inners” approach to some problems of system theory, IEEE Trans. Automatic Control 16:2X3- 240 (1971). E. I. Jury, The theory and applications of the inners, IEEE Trans. Automatic Control 63: 1044- 1068 (1975). J. Maroulas, Ph.D. Thesis, Bradford Univ., England, 1979. J. Maroulas and S. Barnett, Greatest common divisor of generalized polynomials and polynomial matrices, Linear Algebra and A&. 22:195-210 (1978). J. Maroulas and S. Bamett, Some new results on the qualitative theory of generalized polynomials, J. Inst. Math. Appl. 222%70 (1978). J. Maroulas and S. Bamett, Polynomials with respect to a general basis, J. Math. Anal. Appl. 72:177-194 (1979). H. H. Rosenbrock, State Space and Multivariable Thmry, Nelson, 1970. Van Der Waerden, Modan Algebra, Frederic Ungar, New York, 1949. A. I. G. Vardulakis and P. N. R. Stoyle, Generalized resultant theorem, Z. Inst. Math. Appl. 2233-335 (1978).